A multilevel algorithm for the solution of second order elliptic differential equations on sparse grids

Pflaum, Christoph

doi:10.1007/s002110050334

Cited by 18 publications

(26 citation statements)

References 8 publications

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“…This is in contrast to other sparse grid algorithms; see [27, section 2.6]. We also emphasize that the method introduced in [26] does not exploit the product structure (2.9) if present. It thus can naturally cope with PDEs with variable coefficients that do not have such tensor product structure, although, however, only in up to two dimensions.…”

Section: Sparse Grid Discretizationmentioning

confidence: 55%

“…In the context of sparse grids and the hierarchical basis, Downloaded 01/11/16 to 18.51.1.3. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php these prolongation and restriction operators correspond to hierarchization and dehierarchization, respectively [4,21,26].…”

Section: Multigrid and Sparse Gridsmentioning

confidence: 99%

“…We therefore solve our sparse grid system (2.8) with a multigrid method. We specify suitable multigrid ingredients following [26].…”

Section: Multigrid and Sparse Gridsmentioning

confidence: 99%

“…There, the grids are coarsened in each direction separately, i.e., semicoarsening; see Figure 1. We therefore employ the so-called Q-cycle, which traverses all hierarchical subspaces in {H k | k ∈ L} of a sparse grid space by recursively nesting standard V-cycles; see [26] and Figure 2. An algorithmic formulation and an extension to more than two dimensions can be found in [21].…”

Section: Multigrid and Sparse Gridsmentioning

confidence: 99%

“…They are used either as preconditioners [1,11,12,15,16] or within real multigrid schemes [2,3,6,26,33]. We first note that certain problems allow for sparse grid dis-S53 2.1.…”

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confidence: 99%

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A Multigrid Method for Adaptive Sparse Grids

Peherstorfer¹,

Zimmer

Zenger

et al. 2015

SIAM J. Sci. Comput.

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Abstract. Sparse grids have become an important tool to reduce the number of degrees of freedom of discretizations of moderately high-dimensional partial differential equations; however, the reduction in degrees of freedom comes at the cost of an almost dense and unconventionally structured system of linear equations. To guarantee overall efficiency of the sparse grid approach, special linear solvers are required. We present a multigrid method that exploits the sparse grid structure to achieve an optimal runtime that scales linearly with the number of sparse grid points. Our approach is based on a novel decomposition of the right-hand sides of the coarse grid equations that leads to a reformulation in so-called auxiliary coefficients. With these auxiliary coefficients, the right-hand sides can be represented in a nodal point basis on low-dimensional full grids. Our proposed multigrid method directly operates in this auxiliary coefficient representation, circumventing most of the computationally cumbersome sparse grid structure. Numerical results on nonadaptive and spatially adaptive sparse grids confirm that the runtime of our method scales linearly with the number of sparse grid points and they indicate that the obtained convergence factors are bounded independently of the mesh width. 1. Introduction. Many problems in computational science and engineering, e.g., in financial engineering, quantum mechanics, and data-driven prediction, lead to high-dimensional function approximations [23]. We consider here the case where the function is not given explicitly at selected sample points but implicitly as the solution of a high-dimensional partial differential equation (PDE). Standard discretizations on isotropic uniform grids, so-called full grids, become computationally infeasible because the number of degrees of freedom grows exponentially with the dimension. Sparse grid discretizations are a versatile way to circumvent this exponential growth [4]. If the function satisfies certain smoothness assumptions [4, section 3.1] sparse grids reduce the number of degrees of freedom to depend only logarithmically on the dimension; however, the discretization of a PDE on a sparse grid with the (piecewise linear) hierarchical basis results in an almost dense and unconventionally structured system of linear equations. To be computationally efficient solvers must take this structure into account. We develop a multigrid method that exploits the structure of sparse grid discretizations to achieve an optimal runtime that scales linearly with the number of sparse grids points.Multigrid methods have been extensively studied in the context of sparse grids. They are used either as preconditioners [1,11,12,15,16] or within real multigrid schemes [2,3,6,26,33]. We first note that certain problems allow for sparse grid dis-

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Section: Sparse Grid Discretizationmentioning

confidence: 55%

Section: Multigrid and Sparse Gridsmentioning

confidence: 99%

“…We therefore solve our sparse grid system (2.8) with a multigrid method. We specify suitable multigrid ingredients following [26].…”

Section: Multigrid and Sparse Gridsmentioning

confidence: 99%

Section: Multigrid and Sparse Gridsmentioning

confidence: 99%

“…They are used either as preconditioners [1,11,12,15,16] or within real multigrid schemes [2,3,6,26,33]. We first note that certain problems allow for sparse grid dis-S53 2.1.…”

mentioning

confidence: 99%

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A Multigrid Method for Adaptive Sparse Grids

Peherstorfer¹,

Zimmer

Zenger

et al. 2015

SIAM J. Sci. Comput.

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Convergence results for 3D sparse grid approaches

Noordmans

Hemker

1998

Numer. Linear Algebra Appl.

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The convergence behaviour of solution algorithms is investigated for the anisotropic Poisson problem on partially ordered, sparse families of regular grids in 3D. In order to study multilevel techniques on sparse families of grids, first we consider the convergence of a two-level algorithm that applies semi-coarsening successively in each of the coordinate directions. This algorithm shows good convergence, but recursive application of the successive semi-coarsening is not sufficiently efficient. Therefore we introduce another algorithm, which uses collective 3D semi-coarsened coarse grid corrections. The convergence behaviour of this collective version is worse, due to the Jack of correspondence between the solutions on the different grids. By solving for the trivial solution we demonstrate that a good convergence behaviour of the collective version of the algorithm can be retained when the different solutions are sufficiently coherent. In order to solve also non-trivial problems, we develop a defect correction process. This algorithm makes use of hierarchical smoothing in order to deal with the problems related to the Jack of coherence between the solutions on the different grids. Now good convergence rates are obtained also for non-trivial solutions. All convergence results are obtained for two-level processes. The results show convergence rates which are bounded, independent of the discretisation level and of the anisotropy in the problem.

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Efficient Ritz‐Galerkin Discretization of PDEs with Variable Coefficients in arbitrary Dimensions using Sparse Grids

Hartmann

Pflaum

2017

Proc Appl Math and Mech

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Sparse grids can be used to discretize elliptic differential equations of second order on a d-dimensional cube. Using the Ritz-Galerkin discretization, one obtains a linear equation system with O N (log N ) d−1 unknowns. The corresponding discretization error is O N −1 (log N ) d−1 in the H 1 -norm. A major difficulty in using this sparse grid discretization is the complexity of the related stiffness matrix. To reduce the complexity of the sparse grid discretization matrix, we apply prewavelets and a discretization with semi-orthogonality. Furthermore, a recursive algorithm is used, which performs a matrix vector multiplication with the stiffness matrix by O N (log N ) d−1 operations. Simulation results up to level 10 are presented for a 6-dimensional Helmholtz problem with variable coefficients.

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A multilevel algorithm for the solution of second order elliptic differential equations on sparse grids

Cited by 18 publications

References 8 publications

A Multigrid Method for Adaptive Sparse Grids

A Multigrid Method for Adaptive Sparse Grids

Convergence results for 3D sparse grid approaches

Efficient Ritz‐Galerkin Discretization of PDEs with Variable Coefficients in arbitrary Dimensions using Sparse Grids

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